<?xml version="1.0"?>
<div class="add">

<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E2:32_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<a NAME="E3:32_1_1"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_2.html#V2">all-with_const_op</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D3">MSUALG_2:def 3</a><br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:32_1_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="unialg_2.html#D12">UNIALG_2:def 12</a></i>;<br/>
<a NAME="E3:32_1_1_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:32_1_1_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E6:32_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E7:32_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="mssublat.html#E5:32_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_1.html#NM2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="mssublat.html#E2:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="mssublat.html#E5:32_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="mssublat.html#E6:32_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_2.html#D2">MSUALG_2:def 2</a><br/>

<b>thus </b><a NAME="E9:32_1_1_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> 0
<b>by </b><i><a class="txt" href="mssublat.html#E2:32_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="mssublat.html#E5:32_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="mssublat.html#E7:32_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="mssublat.html#T1" target="_self">Th1</a></i>
;<br/>
<b>hence </b><a NAME="E2:32_1_1_1_1"/>
the <a href="msualg_1.html#U2">Arity</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="mssublat.html#E2:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:32_1_1_1"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1_1_2"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:32_1_1_1_2"/>
the <a href="msualg_1.html#U3">ResultSort</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="mssublat.html#E2:32_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:32_1_1"/>
( not  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_1.html#V2">void</a> &amp;  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_1.html#V1">strict</a> &amp;  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_1.html#V6">segmental</a> &amp;  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a> &amp;  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_2.html#V2">all-with_const_op</a> )
 ;<br/>


</div>
